R U : A U VERVIEW OF RABBINIC METHODSu.cs.biu.ac.il/~koppel/rov-traditionversion.pdf · 2006. 9. 29. · Moshe Koppel 1 Moshe Koppel teaches computer science at Bar-Ilan University

Moshe Koppel

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Moshe Koppel teaches computer science at Bar-IlanUniversity.

RESOLVING UNCERTAINTY: A UNIFIEDOVERVIEW OF RABBINIC METHODS

INTRODUCTION

The Talmud frequently discusses situations in which a legal deci-sion must be made in the face of uncertainty with regard to thefacts. Such discussions invoke a number of different principlesbut at least some of these principles are not given explicit formal defini-tion and the relationships between them are not spelled out. In thispaper, I will formally define each of the variety of rabbinic methodsused to resolve (or dispel) empirical uncertainty and place them in thecontext of certain now well-understood probabilistic concepts. In thisway, I hope to present a unified overview of rabbinic laws concerninguncertainty.

The modern theory of probability is twice removed from rabbiniclaws concerning uncertainty. First, in its current form the theory ofprobability is simply the study of a particular class of functions and isnot concerned with assigning probabilities to real-world events. Second,even if on the basis of certain stipulations the theory is applied to actualevents, it remains descriptive and not prescriptive. Nevertheless, certainphilosophical issues which have arisen as a result of attempts to explicatethe meanings of probabilistic statements are highly relevant to a properunderstanding of rabbinic approaches to uncertainty. I will use ideastaken from the study of foundations of probability where these ideasseem helpful but will try to refrain from belaboring the analogy for itsown sake.

One historical point needs to be emphasized. The modern theory ofprobability has its roots in the work of Pascal and others in the seven-teenth century. It would be utterly anachronistic to attribute to Tanna’imand Amora’im any foreknowledge of these developments. Moreover, doingso does not purchase any explanatory power with regard to rabbinic

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approaches to uncertainty. At the same time, the claim that the ancientswere bereft of any systematic thinking with regard to uncertainty isboth arrogant and demonstrably false. I will use modern ideas aboutthe foundations of probability as a starting point for identifying whichprobabilistic insights do and do not lie at the root of rabbinic pro-nouncements on such matters.

Nevertheless, my approach in this article is unabashedly ahistorical:rather than chart a chronological progression of ideas or identify con-flicting schools of thought, I will attempt to harmonize a broad range ofsources. Where a Tanna’itic or Amora’ic source permits multiple interpre-tations, I will not outline all views but rather select the most straightfor-ward or consensual interpretation. Likewise, I will relate to the centralideas discussed in the vast post-talmudic literature devoted to rabbiniclaws concerning uncertainty but, for the sake of offering as straightfor-ward and unified a treatment as possible, I will cite opinions of the com-mentators in an extremely selective manner.1 The fact that I marshal thesupport of a particular commentator regarding a particular point shouldin no way be taken to mean that I can claim such support regardingrelated points.

In the first part of this article, I will use the distinction betweenrubba de-ita kamman and rubba de-leta kamman to motivate a discus-sion of distinct definitions of probability. This will lay the groundworkfor the explication of a number of thorny rabbinic concepts involvinguncertainty and indeterminacy.

INTERPRETATIONS OF PROBABILITY

The Gemara in Hullin 11a-11b interprets the verse “aharei rabbim le-hattot” to mean that decisions of a bet din are decided by majority (to beprecise, a majority of two is required to convict). This is then generalizedto a principle of rubba de-ita kamman (henceforth RDIK; literally: amajority which is in front of us), which includes other cases such as thatof “nine stores,” in which a piece of meat is found in the street and allthat is known is that it comes from one of ten stores, nine of which sellkosher meat. In such cases we apply the principle that kol de-parish me-rubba parish (henceforth parish; literally: that which is removed, wasremoved from the majority). The Gemara states that this inference cov-ers only the principle of RDIK, of which the majority rule in theSanhedrin and “nine stores” are offered as typical examples, but notrubba de-leta kamman (henceforth RDLK; literally: a majority which is


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not in front of us). The Gemara offers a number of examples of RDLKwhere the majority is followed because it would be impossible to per-form mitsvot or adjudicate cases without doing so (but concludes thatprecisely because of that impossibility these cases can’t serve as a basisfrom which to infer a general principle of RDLK). Several cases ofRDLK that are illustrative are that the husband of one’s mother (at thetime of conception) may be presumed to be one’s father, that a childmay be presumed to be potentially fertile and that a murder victim maybe presumed not to have been suffering from a prior life-threateningcondition.

What is the difference between RDIK and RDLK? Although thenames are suggestive, the Gemara offers no explicit definition of RDIKand RDLK and no rationale for treating them differently. We might,however, shed considerable light on the distinction by considering aninteresting philosophical debate dating back to the 1920’s which coverssimilar conceptual territory. The rest of this section will consist of aslightly lengthy diversion through that territory.

Let’s consider carefully what exactly we mean when we say that theprobability of some event is p/q. Early (seventeenth and eighteenth cen-tury) work in probability was motivated to a large extent by games ofchance (coins, cards, dice). Thus when somebody said that “the proba-bility of the event H is p/q,” it was understood that what was meant wasthat the event H obtained in p out of q equally likely possible outcomes.Thus, for example, when we say the probability that the sum of twothrows of a die will be exactly six is 5/36, we mean that there are 36equally likely possible throws and five of them have the desired property.Similarly, in the case of the found meat, there are ten possible sources forthe meat and nine of them are kosher, so we might say that the probabil-ity that the meat is kosher is 9/10. This understanding of probabilisticstatements is usually called the “classical” interpretation.

What is interesting for our purposes is that the classical interpreta-tion turns out to be inadequate as a definition of probability. Thisbecame obvious once insurance companies began using probability the-ory to compute actuarial tables. What does it mean to say that “theprobability that a healthy 40-year-old man will live to the age of 70 isp/q”? What are the q equally likely possible outcomes, p of which findour insuree celebrating his seventieth birthday? No such thing. This ledphilosophers such as Reichenbach and von Mises2 to suggest the “fre-quentist” interpretation of probability: the statement that “the proba-bility that a healthy 40-year-old man will live to the age of 70 is p/q”


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means that of the potentially infinite class of hypothetical healthy 40-year-old men, the proportion who will see 70 is p/q.

It is important to understand that according to each of these inter-pretations, the classical and the frequentist, there is always some subjec-tive aspect in assigning a probability to an event. In the case of classicalprobability this subjective element is rather benign: we need to definethe underlying “equally likely” cases, or what is called in formal parl-ance the “sample space.” For example, in the case of “nine stores” wemight just as plausibly use as our sample space the three shopping mallsin which the stores are concentrated, or perhaps the 10,000 pieces ofmeat that are unequally distributed among the stores. The choice ofwhich sample space is most appropriate is ultimately a matter that mustsimply be stipulated. It is tempting to imagine that the “right” samplespace is the one in which the various elements are equally probable. Butobviously this formulation is circular since it is the very notion of prob-ability that we are trying to define. To be sure, in many cases there is arather obvious first choice of sample space. For example, in tossing adie, we would naturally identify the six possible faces as our samplespace. This intuition rests on some sort of “indifference principle” (whyshould one face be more likely than another?). But such indifferenceprinciples have proved remarkably resistant to precise formulation.Ultimately, the assignment of sample space is a matter of stipulation.

If in the case of classical probability, assigning a probability to anevent requires a bit of judgment, in the case of frequentist probabilitysuch an assignment is fraught with judgment. Think of the example inwhich we wish to determine the probability that a particular child ispotentially fertile (actually in the situation described in the Gemara wewish only to determine that this probability is greater than 1/2). Wewish to do so by invoking some rule that says: there is some referenceclass A in which this child is a member and the expected proportion ofmembers of A which are potentially fertile is p/q. This expected fre-quency is in turn determined by our past experience with members ofclass A and the frequency of fertility they exhibited. But what class A isappropriate? Should A be the class of all young mammals, or all humanchildren, or perhaps the class of all children who share this child’s medicalhistory, or the class of children who share this child’s medical history andgenetic stock? If we define the class too broadly we run the risk that ourexperience with the class is irrelevant to the particular child in question.If we define it too narrowly we run the risk that our experience with theclass is too limited to provide any reliable information with regard to the


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class in general. And if we define it bizarrely (say, the class consisting ofthis child and all major household appliances), the results are, well,bizarre. The selection of the reference class A, as well as the determina-tion that our experience with samples from that class is sufficient to proj-ect some statistical law onto the whole class, are matters of judgment.

Consider now the extreme case of a probabilistic statement such as“the probability that the United States will attack North Korea within twomonths is 60%.” The problem with such statements is that the events inquestion belong to no natural class since the ensemble of relevant factsrenders the case unique. It is implausible that we mean to say that in60% of cases like this an attack occurs, because there aren’t any casesquite “like this.” Since according to the frequentist interpretation everyprobabilistic statement must refer to some class, these statements areutterly meaningless within the frequentist framework and indeed arerejected as such by von Mises and others.

One attempt to salvage such statements as meaningful has involvedyet another interpretation of probability, the “subjectivist” interpreta-tion. According to this interpretation, the statement that the probabilityof some event is p/q is taken to reflect the degree of certainty withwhich some rational observer is convinced of the correctness of thestatement, as might be reflected in a betting strategy. Unlike the previ-ous interpretations, such an interpretation does not require the identifi-cation of any relevant class. For example, for someone to say that theprobability that the United States will attack North Korea within twomonths is 60% is simply to say that they regard as fair either side of abet with 3:2 odds in favor of such an attack occurring.

To summarize, there are at least three different kinds of probabilis-tic statements: classical, frequentist, and subjective.3 For each type, anyinstance of such a statement is meaningful only to the extent that atleast one potentially fuzzy factor can be plausibly defined. In the classi-cal case this factor is a sample set, in the frequentist case it is a referenceclass, and in the subjectivist case it is simply the strength of a hunch.

In the following sections, we will see how various rabbinic methodscan be best understood in relation to these different types of probabilis-tic statements. Moreover, we will see that different ways of resolving thefuzzy aspects of probabilistic statements can neatly account for certainapparent anomalies. In the next section, we will explain differencesbetween the conditions and consequences of RDIK, on the one hand,and those of RDLK, on the other. After that we will clarify when RDIKis applied and when a converse rule (kavu’a) is applied and will eluci-


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date the difference between safek (uncertainty) and indeterminacy.Finally, we will discuss the mechanics of sefek sefeka and contrast it withcases of asymmetric sefekot where the asymmetry is ineffective (en safekmotsi mi-yedei vaddai).

RUBBA DE-ITA KAMMAN AND RUBBA DE-LETA KAMMAN

We will define the principle of RDIK more precisely in the next sectionbut for now it is enough to define it roughly as follows: A randomobject taken from a set a majority of the members of which have prop-erty P, may be presumed to have property P. As so defined, the principledoes not require any (but perhaps the most naive) probabilistic notions.Nevertheless, it is evident that the classical interpretation is fully ade-quate for a probabilistic formulation of RDIK: RDIK amounts to speci-fying the members of the set as a sample space and following the resultwith probability greater than 1/2. Note that RDIK refers specifically toa set of q concrete objects, p of which have some property, while theclassical definition of probability refers more generally to q possible out-comes (which may be abstract).

The classical interpretation is, however, clearly irrelevant to theexamples of RDLK we have seen. The frequentist interpretation, on theother hand, squares with RDLK perfectly.4 Simply put, all examples ofRDLK are statistical laws: most children born to married women arefathered by their husbands, most children are ultimately fertile, mostpeople are not about to die, etc.

The identification of RDIK with the classical interpretation andRDLK with the frequentist interpretation will help us clear up a num-ber of difficulties.5 Let us begin with the question of which is stronger,RDLK or RDIK. Aharonim have marshaled proofs for each possibility,the most salient of which follow.

The strength of RDLK relative to RDIK can be clearly seen in thefollowing: It is well established that we don’t convict in capital casesbased on mere likelihood (Sanhedrin 38a). Thus, consider the case ofan abandoned baby boy, called an asufi, whose mother is one of a givenlarge set of women one of whom is a non-Jew. In this case, there is aRDIK in favor of the child’s Jewish maternity. While such a child maybe regarded as a Jew for certain purposes, a woman who eventuallymarries him cannot be convicted of adultery, “she-en horgin al ha-safek”(Rambam, Issurei Bi’a 15:27; see also Makhshirin 2:7, Ketubot 15a).


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Nevertheless, consider another case of uncertain maternity, in which awoman has a relationship with a child that is typical of that of motherand son but, as is generally the case, there are no witnesses to the birth.In this case, there is a RDLK in favor of the woman’s maternity. If sheand the “son” are witnessed having sexual relations, they can be con-victed for incest “she-sokelin ve-sorefin al ha-hazaka” (Kiddushin 80a,Rambam, Issurei Bi’a 1:20). Clearly, RDLK in these cases is strongerthan RDIK.

There are other cases, however, in which RDIK appears to bestronger than RDLK. For example, R. Meir holds that haishinan le-mi’uta—a rov does not trump a contrary hazaka de-me-ikara unless it isan overwhelming rov (Yevamot 119b). Thus, for example, dough ofteruma that was last known to be tahor but was found in the proximityof a child who is tamei cannot be burned, according to R. Meir, on thebasis of a RDLK that children typically pick at dough in their vicinity(Kiddushin 80a). Tosafot (Yevamot 67b, s.v. en hosheshin; Yevamot 119a,s.v. kegon; see also Mordekhai on Hullin, ha-Zero’a 737) argue that thisprinciple holds only with regard to RDLK but RDIK always trumps ahazaka de-me-ikara. Moreover, according to R. Yohanan, in the case ofthe teruma dough even the rabbis who disagree with R. Meir wouldconcede that the teruma can’t be burned on the basis of this RDLK.Nevertheless, they would not so concede in a case of RDIK (seeKiddushin 80a, Rashi s.v. im rov). Thus, in these cases RDLK is weakerthan RDIK.6

We might be able to reach a definitive answer regarding which isstronger, RDIK or RDLK, by explaining away one or the other set ofproofs. But to do so would be to answer the wrong question. Tounderstand the crucial difference between RDIK and RDLK, let’s recallthe difference between the classical interpretation of probability and thefrequentist interpretation.

In the case of classical probability, the part that is left to judgment israther limited. Typically, a rather straightforward sample space is takenfor granted. Once that’s taken care of, assigning a probability is a simplematter of calculation. (In fact, in the limited case of RDIK, the casesneed only be counted.) In the case of frequentist probability, however,selecting a reference class and then estimating frequencies within theclass requires a substantial investment of judgment. With what confi-dence can we assert that for some class A the event in question occurswith some sufficiently high frequency? Answering this question, evenloosely, is inevitably a matter of judgment.


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The crucial difference, then, between RDIK and RDLK is that,while RDIK is a counting principle that can be applied on an ad hocbasis, every RDLK is a general statistical law that can only be applied if ithas received rabbinic sanction. Since RDLK is always a product of rab-binic judgment, it stands to reason that this judgment is exercised vari-ably. The apparent contradiction regarding the relative strengths ofRDIK and RDLK simply reflects the fact that different applications ofRDLK are assigned different strengths (both in terms of the strengths ofthe laws themselves and in terms of the strength of the evidence for thelaws). On the other hand, applications of RDIK, which do not—so tospeak—pass through rabbinic hands, are all treated in a uniform manner.

Consequently, if you’ve seen one RDIK you’ve seen them all.Unless there is some countervailing principle which prevents its applica-tion, RDIK is a decision procedure which resolves, but does not dispel,uncertainty in favor of the majority regardless of whether p/q is .99 or.51. That is, in applying the principle of RDIK we acknowledge thatthere is uncertainty but the RDIK allows us to decide in favor of themajority much in the way that a majority vote settles a case in court.Invoking RDIK is not sufficient, however, to achieve the degree of cer-tainty necessary to establish the facts of a capital case.

Unlike RDIK, however, there are various types of RDLK. There arethree types of decision rules and, depending on rabbinic judgment,RDLK can be any one of them.

—The middle type is the one we have seen in the case ofRDIK—a resolution procedure. These are often referred to as“hakhra’a.”7 An example of an RDLK of this type is that mostbirths are not of healthy males (Hullin 77b).

—There are stronger decision rules which simply render irrele-vant the minority possibility—some examples of RDLK are treat-ed as certainties in the sense that we proceed as if the uncertaintyhas not simply been resolved but rather has been dispelled alto-gether (or as some would have it, there is no ledat ha-safek).These are often referred to as “berur.” It is about these that wesay “sokelin al ha-hazakot”—in capital cases certainty is requiredand these examples of RDLK, unlike any example of RDIK, doindeed provide certainty for legal purposes.8

—Finally, there are weaker decision rules which are merely“defaults” in the sense that they are applied only as last-resort tie-


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breakers when no more substantive decision rule is available.These are often referred to as “hanhaga.” The typical example ofa default rule in halakha is hazaka de-me-ikara (with the possibleexception of hezkat mamon—possession—which is regarded as asubstantive desideratum and not a mere default).9 In some cases,RDLK is established merely as a default rule so that at most it canneutralize,10 but not defeat, another default rule such as hazakade-me-ikara. For R. Meir, most cases of RDLK are of this variety.

RUBBA DE-ITA KAMMAN AND KAVU’A

Let us now return to the principle of RDIK and attempt to define itmore precisely. We have already seen that according to the Gemara(Hullin 11a), this principle covers both the case of majority vote in theSanhedrin and that of “nine stores” where the meat is found on thestreet. Moreover, the Gemara often invokes the related, though clearlynot identical, principle of bittul be-rov: a mixture of permitted and for-bidden objects may sometimes be assigned the status of the majority.Although the Gemara does not specify the source of this principle,most commentators follow the opinion of Rashi (Gittin 54b, s.v. loya’alu) that it is derived from “aharei rabbim le-hattot” as well.

The generalization from the case of majority vote to cases such as“nine stores” is not inevitable—the case of voting involves legal, notempirical uncertainty. As R. Elchonon Wasserman (Kunteres DivreiSoferim 5:7) puts it: if Eliyahu ha-Navi declared the questionable pieceof meat to have come from the minority we could take his word for it,but if he ruled in accord with the minority position in the Sanhedrin wewould ignore him (as in the case of tanuro shel akhnai in Bava Metsia59b). Similarly, it has been argued (Sha’arei Yosher 3:4) that the exten-sion to bittul be-rov does not seem, on the face of it, to be one that canbe glibly asserted. Clearly, the principle the Gemara wishes to base onaharei rabbim le-hattot is sufficiently general that it covers all of theabove cases.

Before we consider what this principle might be, let’s consider theremarkably similar situation with regard to another decision principle,namely, kol kavu’a ke-mehetsa al mehetsa dami (henceforth kavu’a; literal-ly: that which is fixed is as half and half). Like RDIK, the case identified inthe Gemara as the “source” case of kavu’a is a technical, legal question.Someone throws a stone into an assembly of nine Israelites and one


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Canaanite, intending to kill whichever person the stone happens to hit.The question is whether this unspecific intention is sufficient intention tokill an Israelite to warrant conviction for murder. The Rabbis apply theprinciple of kavu’a to determine that the Israelite majority does not ren-der the intention sufficient (Ketubot 15a). What exactly the principlemight be requires explanation. But note that in this case there is no doubtthat the actual victim was indeed an Israelite and not a Canaanite. Theissue under discussion is only whether the intention to kill “some memberof this group” can be regarded as the intention to kill an Israelite. Thus,there is no uncertainty regarding any of the facts of this case and no deci-sion-method for resolving empirical uncertainty is called for.

The Gemara then cites as the classic example of kavu’a, the parallelcase to that of “nine stores” that we considered above: “If there arenine stores which sell kosher meat and one which sells non-kosher meatand someone took [meat] from one of them but he doesn’t know fromwhich one he took, the meat is forbidden.”

The parallelism between RDIK and kavu’a is remarkable. In both,the “source” case involves court procedures and includes no elementsof actual uncertainty, and in both the standard case is a version of “ninestores,” in which the central issue is apparently one of uncertainty. Thissuggests that RDIK and kavu’a do not directly concern uncertainty, butrather are dual principles regarding mixed sets that cover cases of uncer-tainty as a by-product.

The principle of RDIK might thus be formulated this way:

Given a set of objects the majority of which have the property Pand the rest of which have the property not-P, we may, under cer-tain circumstances, regard the set itself and/or any object in theset as having property P.

The principle of kavu’a is the opposite of this:

Given a set of objects some of which have the property P and therest of which have the property not-P, we may, under certain cir-cumstances, regard the set itself, and consequently any object inthe set, as being neither P nor not-P but rather a third status. Wecan call this status hybrid, or perhaps, indeterminate.

It is important to note that RDIK comes in two varieties: RDIK canassign a single status to the entire mixed set (as in the case of bittul) orit might assign a status directly to an individual object in the set (as inparish). Kavu’a, on the other hand, comes in only one variety: a hybrid


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status must be assigned to a set and then only indirectly to an individualitem in the set. When kavu’a is invoked, each individual item in the setloses its individual identity and is regarded simply as a fragment of anirreducibly mixed entity. It is not treated as an individual of uncertainstatus but rather as a part of a set that is certainly mixed.

Given this, we are ready to answer the central question: When dowe apply RDIK and when do we apply kavu’a?

Roughly speaking, the idea is that when an object is being judged inisolation, it must be assigned a status appropriate to an individual object;when it is judged only as part of a set, it can be assigned some new statusappropriate for a set. Kavu’a can only be invoked in the latter case. Tosee this distinction very starkly, consider two scenarios in each of whichwe have before us a box containing nine white balls and one black ball.

Scenario 1: I reach into the box, pull out one ball without show-ing it to you and ask: What is the color of this ball?

Scenario 2: I don’t reach into the box, but instead ask: What is thecolor of a random ball in this box?

In the first case, if you were to answer, say, “black,” your answerwould be either true or false, but either way would be an appropriateresponse to the question that was asked. There is a determinate answerto the question, although this answer is unknown to you. In the secondcase, the answer “black” (or “white”) is neither true nor false, sincethere is no determinate answer to the question. You could say nothingmore specific than that the box contains both white and black balls.

Obviously, the case of the stone-thrower considered above is analo-gous to scenario 2—asking about the status of an unspecified memberof the group is like asking about the color of an unspecified ball. Theappropriate level at which to assign status in this case is the level of theset, not the level of the individual, and the set is indeed mixed. This isthe sort of case in which kavu’a can be invoked.

By contrast, a piece of meat that is found in the street is clearlyanalogous to scenario 1—the status of a particular item is in question.This is the kind of case in which RDIK is invoked. Now admittedly, thecase of a piece of meat bought in one of the stores might plausibly beregarded as analogous to scenario 1 since the act of buying could beconsidered analogous to pulling out a specific ball. However, the rab-binic principle is, somewhat counter-intuitively, otherwise: apparently,the critical moment is the one prior to actually encountering the piece


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in question. When the piece is found on the street, it is judged as anindividual because prior to the moment that it is found, it is already nolonger “in the set.” When the piece of meat in question is bought inthe store, prior to its being bought it is indeed “in the set.”

The distinction between kavu’a and RDIK might be restated in termsof the issue of sample space selection considered above. RDIK assumesthe “standard” sample space. In the case of the meat found in the street,that sample space is the set of stores. But kavu’a entails the selection of anon-standard, but entirely sensible, sample space: the single element con-sisting of the entire set of stores. This single item is mixed.

Let us now spell out in detail the precise method for determiningwhen to apply RDIK and when to apply kavu’a.

1. First, there are a number of cases in which kavu’a cannot beinvoked because a hybrid status is inappropriate.

—In the case of a vote in Sanhedrin which is, by definition, amechanism for rendering a decision.

—If uncertainty regarding the status of an individual object whichbelonged to the set arose only after the object had been isolatedfrom the set (parish), then it is this object alone which must beassigned some status. While a member of a set consisting of objectssome of which are P and some of which are not-P can be assigned ahybrid status as part of the set, an individual object being assigneda status on its own cannot. Thus, we need to choose either P ornot-P for this object and we choose the majority of the set fromwhich it comes. For example, in the case of “nine stores” in whichthe meat is found on the street, the isolated piece of meat isassigned either the status “kosher” or the status “non-kosher.”

—Similarly, if the set is somehow “incohesive,” so that each objectin it is regarded as having left the set, we apply RDIK and notkavu’a. Thus, for example, a set of travelers passing through atown do not constitute a set for purposes of kavu’a, while the resi-dents of the town do (Ketubot 15b; and compare Yoma 84b).11

—Finally, if it is not certain that the set contains any objects thatare, say, not-P, then the set is said not to satisfy the condition ofithazek issura. Such a set cannot be assigned a hybrid status andRDIK is invoked rather than kavu’a. Thus, the Tosefta (Taharot6:3) already considers a case in which we are given a mixture of


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ten loaves, including one loaf that is tamei, that is eaten in tworounds of five loaves each. Those who eat in the first round aretameh because at that point the set certainly contains one tameiloaf, but those in the second round are tahor because by then theset might not contain a tamei loaf. Similarly, a mixture of objectsincluding one that has been used for idolatry is forbidden for usebut if one of the objects is destroyed the mixture is permitted foruse (Zevahim 74a as understood by Rambam, Avoda Zara 7:10).12

To summarize: in all cases in which we are not assigning a status toa mixed set, kavu’a is not invoked but rather RDIK.13 Note thatalthough in these cases the membership of the doubtful item in the set,or the cohesiveness of the set itself, may be inadequate for invokingkavu’a, this does not diminish the relevance of the set for purposes ofRDIK. Thus, for example, even though the piece of meat found on thestreet cannot be assigned a hybrid status because it is not part of the set,the fact that the meat is known to have originated in the set still rendersthe composition of the set (i.e., the majority) relevant to determiningthe status of the piece.

2. When the above rule does not apply, so that at issue is the sta-tus of a mixed set, we first check if the principle of bittul can be applied.If bittul can be applied, the set is no longer regarded as a mixed set butrather as a uniform set, and kavu’a is inapplicable. Thus, for example, amixed set consisting of one (non-distinguishable) non-kosher piece ofmeat and more than one kosher pieces is regarded as a permissible set.14

3. Finally, there are a number of cases in which we are dealingwith a mixed set but bittul is not applicable:

—First, if the objects in the set are each identifiable as either P ornot-P (nikkar bimkomo). For example, in “nine stores” the statusof each store is known, it is only the origin of a particular piece ofmeat that is in doubt. Clearly, in such a case, we can’t define theset as either P or as not-P; as a set it is both.

—Second, if individual objects in the set are each regarded as suf-ficiently significant that the status of each cannot be subordinatedto the status of the set (hashivi ve-lo beteli) or if bittul is inapplica-ble for any other reason. Thus, given a herd of oxen including onethat has been sentenced to death and is forbidden for use, wecan’t invoke bittul due to the significance of living creatures, and


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hence we invoke kavu’a by default (Zevahim 73b).

—Third, if the set includes an equal number of objects that are Pas are not-P. In such a case bittul is obviously not possible.

In each of these cases15 we are dealing with an irreducibly mixed setand the principle of kavu’a is invoked: the set is assigned a new hybridstatus (P and not-P) as are individual objects drawn from the set.

KAVU’A AND SAFEK

The crucial distinction between safek, which reflects uncertainty regard-ing an individual object, and kavu’a, which is a definite hybrid statusassigned to a set, cannot be over-emphasized. When kavu’a is invoked,it is the definite mixed status of the entire set that concerns us and notthe uncertain status of any individual item in the set. It is generally thefailure to appreciate this distinction which leads to the conclusion thatkavu’a is completely counter-intuitive.

Let’s consider for a moment the alternative, more common, expli-cation of kavu’a as merely a leveling of the playing field in which thecase is treated as a symmetric safek. On this understanding, which Ireject, the sample space would contain two elements: kosher and non-kosher. According to my explanation, in cases of kavu’a, the samplespace consists of a single element: the entire mixed set. Might not thephrase mehetsa al mehetsa suggest that the rule is in fact that we assigneach status a probability of 1/2, that is, that we have a sample spaceconsisting of two elements? Why do I reject this possibility?

First of all, because such a rule would be arbitrary and the one Iargue for is perfectly sensible. Moreover, the notion that mehetsa almehetsa refers to a probability of 1/2 is utterly anachronistic. Theassignment of probabilities to the range [0,1], so that 1/2 is in themiddle, is a relatively recent convention. The phrase mehetsa al mehetsacertainly refers to set composition and not to probability. Specifically, itrefers to the third case in Rule 3 of the kavu’a/RDIK rules above inwhich kavu’a applies to a mixed set that includes an equal number ofobjects that are P as are not-P. The point of the phrase kol kavu’a ke-mehetsa al mehetsa dami is that in all cases that satisfy the conditions forkavu’a, RDIK is not invoked just as it is obviously not invoked in thecase where there is no majority.

Finally, there are important halakhic differences between cases that


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are deemed safek and cases where kavu’a is applied. For example, if aperson had before him two indistinguishable pieces of meat, one kosherand one non-kosher—a case of kavu’a—and he ate one of them, he isobligated to bring an asham taluy. But if he had before him one piece,possibly kosher but possibly non-kosher—a case of safek—he is not soobligated (Rambam, Shegagot 8:2, based on Keritut 17b).16 Similarly, ifa mouse takes a piece from a mixed pile of pieces of hamets and ofmatsa, in a manner such that the principle of kavu’a would apply, into ahouse which has been inspected for Pesah, the house must be re-inspected. But if it took a single piece of which has an even chance ofbeing hamets or matsa into the house—this is a safek—the house neednot be re-inspected (Pesahim 9a as understood by Rambam, Hamets u-Matsa 2:10-11). In the case of safek, we can presume that an inspectedhouse remains free of hamets since one possible resolution of the uncer-tainty regarding the subsequent events is consistent with this presump-tion. In the case of kavu’a, however, there is no uncertainty to resolve.Rather, some object of known mixed status has certainly been broughtinto the house; this is enough to nullify the presumption.

Now that we have established that cases of kavu’a are not cases ofsafek, which cases are in fact safek? The status of an object is safek whenit is not judged as part of a set (so that kavu’a and bittul do not apply)and it has not been removed from a set with a majority (so that parishdoes not apply)17 and it does not belong to some reference class forwhich some statistical law is known (so that RDLK does not apply). Aclean example of safek is one in which a piece of meat is found in thestreet and might have come from one of two stores, one kosher and onenon-kosher.

Actually, few cases of safek are that neatly symmetric. In the follow-ing sections, we will deal with two types of asymmetric safek, safek ha-ragil and sefek sefeka, and see which types of asymmetries matter andwhich don’t.

EN SAFEK MOTSI MI-YEDEI VADDAI

In several places in the Gemara we find the principle en safek motsi mi-yedei vaddai. For example, in Avoda Zara 41b, Resh Lakish argues thatif an idol is found broken we can assume that its owner renounces itand thus it is no longer forbidden to make use of it. R. Yohanan rejectsthis argument on the grounds that it was certainly (vaddai) initially an


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TRADITION

idol but only possibly (safek) renounced and thus on the basis of theprinciple en safek motsi mi-yedei vaddai we may not use it. Tosafot (s.v.ve-en safek; Hullin 10a, s.v. taval ve-ala) points out that the safekreferred to in such cases is in fact a “safek ha-ragil”—that is, it is morelikely than not that the idol was renounced. According to Tosafot, then,at least some cases of en safek motsi mi-yedei vaddai are actually cases ofa majority failing against a hazaka de-me-ikara. But then it wouldappear that these cases contradict the widely accepted rule (against theview of R. Meir) that a majority defeats hazaka de-me-ikara (Nidda18b). Why are these cases treated differently?

We can answer this question by answering a more elementary one:what exactly is a safek ha-ragil and why is it not simply called rov? R.Osher Weiss (Minhat Osher, Bereshit 58) suggests that what Tosafot calla safek ha-ragil is specifically a case in which some event does notbelong to any recognized reference class covered by a statistical law. Insuch cases, for which von Mises would argue that the notion of proba-bility is undefined, the principle of RDLK does not apply. Rather, thesafek is ragil only in the sense that a typical observer might find onepossibility to be subjectively more likely than not. But this subjectiveprobability is not relevant; a safek ha-ragil is, for legal purposes, no kindof majority at all.18

Simply put, the term safek refers not to cases where two possibilitiesare known to be equiprobable, but rather to cases in which two possibili-ties exist and neither RDIK nor RDLK are applicable. We might think ofa safek as a case with a sample space consisting of two opposite elements.

SEFEK SEFEKA

Not every case in which both RDIK and RDLK fail necessarily results intwo possibilities. Recall that in our comparison of RDIK with the classi-cal interpretation of probability above, there was a glaring gap. Thesample spaces in RDIK were limited to concrete objects in a set. Whathappens when an object is not drawn from a set of objects in some pro-portion (so that RDIK in the usual sense does not apply), but there aremore than two possibilities for assigning its status. Does there existsome extension of RDIK to such abstract sample spaces? According toone opinion, sefek sefeka is just such an extension.

The principle of sefek sefeka is this: If a particular prohibition holdsonly if both conditions A and B hold, and in fact both A and B are indoubt, then we can assume that the prohibition does not hold. Rashba


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in his Responsa (1:401) states that sefek sefeka operates “like a rov.”There are two main branches of thought regarding the nature of thisanalogy. According to one approach, sefek sefeka is like rov in its effectbut not in its mechanics. Thus according to Ra’ah (Bedek ha-Bayyit onTorat ha-Bayyit 4:2) the principle of sefek sefeka may simply arise natu-rally from the iterative application of the rules for handling a singlesafek; the first safek reduces the issur to a de-rabbanan and the secondrenders it permitted (see also Sha’arei Yosher 1:19). A related interpreta-tion, ascribed to R. Yosef Dov Soloveichik, is that we regard a safekbetween permitted and only possibly forbidden as if there is no ledatha-safek. Consequently, even in instances of sefek sefeka in which itmight be possible to ascertain the actual facts, one is not obliged tomake a special effort to do so (Rosh on Avoda Zara, En Ma’amidin,35, and Rema, Yoreh De’a 110:9). Understood this way, sefek sefeka hasthe effect of one type of RDLK: in each, there is no ledat ha-safek.

A second approach to sefek sefeka, the one of primary interest forour purposes, extends the analogy to rov to include the mechanismthrough which sefek sefeka works. Thus R. Y. S. Natanzon (ResponsaSho’el u-Meshiv 1:196 and many other places) elaborates that sefek sefekacan be thought of as an extension of the idea of RDIK in which thesample space consists of the set of possible truth assignments to the var-ious individual conditions.19 For example, a newlywed woman who isfound to have had sexual relations prior to marriage would be forbid-den to her husband if she had done so (a) subsequent to betrothal, and(b) consensually (Ketubot 9a). Of the four possibilities (subsequent/consensual, subsequent/non-consensual, prior/consensual, prior/non-consensual), only one renders her forbidden. Consequently, if we are indoubt as to which of these four possibilities is indeed the case, we canfollow the majority and she would not be forbidden to her husband.

If indeed sefek sefeka is a type of rov, in its mechanics as well as itseffect, it is certainly not RDLK since many examples of sefek sefeka, suchas the one above, are not amenable to explanation in terms of RDLK.20

Moreover, there are differences between sefek sefeka and RDLK even interms of effect. We have already seen that R. Meir’s principle of haishi-nan le-mi’uta applies only to RDLK and not to RDIK. But in Nidda59b we find that (according to R. Yohanan) R. Meir invokes a sefek sefe-ka against hazaka de-me-ikara and appears unconcerned with the prin-ciple of haishinan le-mi’uta. This is consistent with sefek sefeka as anextension of RDIK but not as RDLK.21

The most intriguing aspect of the identification of sefek sefeka with


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TRADITION

RDIK is the consequent link to the classical interpretation of probabili-ty. The issues that are raised by the commentators regarding restrictionsto the application of sefek sefeka are remarkably similar to those subse-quently raised with regard to the classical interpretation of probability.Consider, for example, the statement that “the probability that the sumof two throws of a die will be exactly six is 5/36.” The classical inter-pretation of this statement is that in a sample space consisting of 36possible throws of the dice, five have the desired condition. But, as wehave seen earlier, this choice of sample space is not inevitable; it must bestipulated. Admittedly, this particular sample space is intuitively sensiblebased on two hard-to-pin-down “indifference principles.”

—P1. For each toss, the six possibilities are plausibly symmetric.

—P2. The two tosses are plausibly independent—that is, the possi-bilities for each toss remain plausibly symmetric regardless of theresult of the other toss.

But critics of the classical interpretation have pointed out that anyattempt to regard the choice of a sample space based on such principlesas anything more than mere stipulations inevitably runs aground on atleast one of the following two problems:

—Q1. Primitive claims of plausible symmetry must not be con-fused with precise claims of equiprobability. To do so would be toinvite infinite regress by introducing probability into the definitionof probability.

—Q2. The same problem sometimes permits different plausiblesymmetries that may lead to contradictory conclusions.22

Consider now the central issues regarding the applicability of sefeksefeka. Recall that the idea is that both A and B are necessary conditionsand both are in doubt. Rishonim have limited the applicability of sefeksefeka to cases in which two “indifference principles” can be invoked:

—P1’. A and not-A, and B and not-B, respectively, are plausiblysymmetric in the sense that rov can’t be applied to either one ofthem (Ketubot 9a, Tosafot s.v. ve-i ba-it).

—P2’. A and B must be independent (“sefek sefeka ha-mitha-pechet”).23

Thus, for example, if there were a firm rule that most cases of sexual


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relations are consensual, our newlyweds’ marriage above could not besaved by sefek sefeka. Moreover, the sefek sefeka works only because thequestion of consent is regarded as independent of the question of tim-ing (prior to or subsequent to betrothal), in the sense that resolvingone question would not resolve the other.

In fact, Rishonim are at pains to point out that the plausible sym-metries underlying the definition of sefek sefeka must be regarded asmere rabbinic stipulations:

—Q1’. Rivash (Responsum 372) points out that symmetry in thesecases should not be confused with equiprobability.

—Q2’. Tosafot note that by formulating the problem differentlywe might find plausible symmetry between A&B and not-(A&B)thus destroying the sefek sefeka (e.g. “shem ones had hu,” Ketubot9a, Tosafot s.v. ve-i ba’it).

Continuing with our example, no claim is made that the probabilitythat relations were consensual is precisely _. It is enough that there aretwo possibilities, consensual or non-consensual, and no firm rule ren-dering one more likely. Moreover, as Tosafot notes, the possibility thatthe relations were non-consensual could in principle be counted as mul-tiple possibilities by distinguishing violent rape from statutory rape. It isa matter of stipulation that we do not do so but rather bundle all casesof rape under a single label.

In short, attempts to explicate sefek sefeka, like attempts to explicatethe classical interpretation of probability, contend with the need forsome fuzzy symmetry criterion, more primitive than equiprobability, onthe basis of which to choose a sample space.24 This parallelism makes ittempting to see in sefek sefeka precisely the kind of extension of RDIKrequired to complete the analogy with the classical interpretation ofprobability.

Since very few cases of sefek sefeka are actually considered in theGemara, it is unclear how broadly we might apply it. In the most restric-tive interpretation, sefek sefeka would only apply where two conditionsare required and each condition permits precisely two possibilities: yes orno. All the cases in the Gemara are of this type.25 In a broader interpre-tation, the breakdown to independent conditions might just be a short-cut for counting elementary cases to determine if a majority of them per-mit or forbid. For example, suppose widgets came in three colors (red,white, blue), and three sizes (small, medium, large), and that a widget is


20

TRADITION

forbidden precisely if it is red OR blue AND medium OR large. Then,by the broader interpretation, a particular widget of unknown color andsize might be permitted since only 4 of 9 possibilities are forbidden. By anarrower interpretation, uncertainty regarding color and uncertaintyregarding size would each be resolved independently in favor of a 2 of 3majority to forbid and the widget would be forbidden.

According to the broader interpretation, constraint P1’ above shouldbe interpreted as strictly analogous to P1: for purposes of case counting,the cases must be plausibly symmetric but there need not necessarily beprecisely two possibilities within each constituent safek. Note that evenaccording to this broad interpretation, sefek sefeka does not provide ablanket license for multiplying numerical probabilities. It merely offers ashortcut for counting cases. As explained at great length by the Shakh(Yoreh De’a 110:9, Dinei Sefek Sefeka), where case counting is inapplica-ble, such as when the constituent sefekot involve conceptually incommen-surable aspects of the situation, sefek sefeka does not apply.

SUMMARY

To summarize, rabbinic methods for resolving uncertainty regardingthe status (P or not-P) of some object are non-numerical but ratheryield one of the following outputs:

—certainly P / not-P—probably P / not-P—uncertain (safek)—hybrid (mehetsa al mehetsa )

These methods can be roughly summarized by the following proce-dure:

1. If the object is known to belong to some fixed set containingobjects that are P and others that are not-P, then use the rulesoutlined above to apply either RDIK or kavu’a. When RDIK isinvoked, the conclusion is either probably P or probably not-P.When kavu’a is invoked, the conclusion is mehetsa al mehetsa.

2. If the object is not known to belong to some fixed set, but doesbelong to some class about which we have some recognized statis-tical law (RDLK) concerning the property P, then assign its statusaccording to that law. Depending on the strength assigned to thelaw, this status might be regarded as certain or as merely probable.


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3. When the necessary conditions for P or not-P to hold can,according to rabbinic judgment, be most naturally formulated interms of a multiplicity of plausibly independent symmetric sefekot,we resolve in accord with the majority of theoretically possibleinstances (sefek sefeka). This conclusion is regarded as probable.

4. If none of the above allows resolution, the status of the objectis safek. In such cases, second-order default rules might be invokedto determine a course of action. These second-order rules involvethe nature (tum’a, yuhesin, mamanot, or issurin) and severity (de-oraita or de-rabbanan) of the prohibition in question and variouspresumptions (hazakot), a detailed discussion of which is beyondthe scope of this article.26

NOTES

1. This literature includes a number of classical book-length treatments, suchas Shev Shema’at’ta, Sha’arei Torah, and Sha’arei Yosher. In addition, PeneiYehoshua often makes reference to an unpublished monograph he wrote onthis topic called Kelal Gadol. Some recent contributions to the classical lit-erature that are especially noteworthy can be found in the writings of R.Osher Weiss. See especially his Minhat Osher on Bereshit, Chapter 58 andShi’ur on Parashat Mishpatim (5763, transcript distributed by MakhonMinhat Osher). An important contemporary book that treats this topic froma historical perspective and covers a number of sources discussed in thispaper in detail is Nachum L. Rabinovitch, Probability and StatisticalInference in Ancient and Medieval Jewish Literature (University of Toronto,1973). Contemporary articles devoted to probabilistic aspects of rabbinicmethods for handling uncertainty are too numerous to list. Three articleswhich I have found exceptionally useful appeared in Higayon (Vol. 4): L.Moscovitz, “On the Principles of Majority and Ithazek Issura” (Hebrew);N. Taylor “The Definition of Rov in Halakha” (Hebrew); Y. Werblowsky,“Rov and Probability.” My gratitude to all these authors. Two importantpapers by J. Beck and V. Shtern, “The Talmudic Concepts of Safek andSefek Sefeka” and “The Talmudic Concept of Zil Batar Ruba,” which Ibecame aware of only as this article was completed, appeared in OtambeMoadam (Bulletin of the Young Israel of Brookline), Nissan 5761 andTishrei 5762, respectively. Finally, I am grateful to a good number of friendsand colleagues who made very helpful comments on earlier drafts of thisarticle. I am not listing them by name only for fear of inadvertent omission.

2. See R. von Mises, Probability, Statistics and Truth (Dover, 1957), originallypublished in German (Springer, 1928).

3. To be sure, these three do not constitute an exhaustive list of all interpreta-tions that have been suggested. Other interpretations, such as the “logical”


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TRADITION

interpretation, purport to subsume one or more of these. Certainly, for ourpurposes these three will suffice.

4. See the discussion in Rabinovitch, Chapter 3.5. The case should not be overstated. Certainly the rabbis regarded majority

as relevant for resolving uncertainty and certainly they distinguishedbetween two distinct kinds of majority. While it is true that these notionsof majority can be neatly embedded in full-blown theories of numericallyquantifiable probability, it certainly does not follow that the rabbis were inconscious possession of any such theory. Nor, by the way, is this necessarilya bad thing. Although scholars since Leibniz have occasionally toyed withthe idea of using probability theory to numerically quantify evidence forlegal purposes, one is hard-pressed to find even a single instance in whichsuch flights of fancy have advanced the cause of justice. See, for example,the epilogue of J. Franklin, The Science of Conjecture (Johns HopkinsUniversity, 2001).

6. It has also been suggested that the principle en holkhin be-mamon ahar ha-rov holds only with regard to RDLK but that RDIK does apply even fordinei mamonot (Rashbam, Bava Batra 93a s.v. de-hu gufeh; Terumat ha-Deshen 314), but it is not clear that this distinction is relevant here. Seefootnote 9 below.

7. The terms I use here, “hakhra’a,” “berur,” and “hanhaga,” are those usedin the lengthy discussion in Sha’arei Yosher 3:1-4. Although the distinc-tions among these three categories are implicit already in the Gemara,explicit conceptualization of them is a relatively recent phenomenon. As aresult, there is no consensus with regard to nomenclature. Often, differentnomenclature is employed, sometimes even with some of these same termsused in different permutations. R. Hayyim Volozhin (cited in ResponsaNahalat David, 24) notes that the word “hazaka” is used in all three ways.

8. To be sure, in capital cases, the act which is grounds for conviction mustbe established by direct witnesses and no amount of circumstantial evi-dence is sufficient (see Sefer ha-Mitsvot, Lo Ta’aseh 290). Even thestrongest type of RDLK is applied only towards establishing the necessarybackground facts (see Shev Shema’at’ta 4:8). It should also be noted thatlegal certainty is required only with regard to the facts of the case; obvious-ly with regard to the procedural matter of voting for conviction, a proce-dural majority—i.e., RDIK—is sufficient.

9. This is the most plausible explanation of Shemuel’s oft-cited principle enholkhin be-mamon ahar ha-rov. In fact, though, the Gemara applies thisprinciple only twice (Bava Kamma 27a and Bava Batra 92b) and boththose cases permit a much narrower interpretation of the principle, namely,that typical market behavior is irrelevant for establishing the intentions ofthe participants in a specific transaction.

Note also that there is at least one case (Ketubot 75b) in which R.Gamliel appears to hold that hezkat ha-guf trumps hezkat mamon. PeneiYehoshua argues that according to this view hezkat ha-guf is also more thana default rule, but other explanations are possible. The case of hezkat ha-guf is in any case interesting in a way that bears mentioning: unlike otherforms of hazaka de-me-ikara, which are purely normative principles, the


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principle of hezkat ha-guf is rooted in a particular model of the real world,namely, one in which as many facts as possible persist as long as they arenot known to have been contradicted. Researchers in artificial intelligencefind such models to be especially useful for many technical reasons.Compare, for example, Y. Shoham, Reasoning About Change (MIT Press,1988), Chapter 5 and R. Elchonon Wasserman, Kovets Sh’iurim, Ketubot75b, Paragraph 265.

10. To be precise, the duel between such a rov and the competing hazaka de-me-ikara results in a standoff. See Yevamot 119a and Kiddushin 80aTosafot s.v. semokh. Later we will encounter cases in which a purported rovis so dubious that it is simply defeated by a hazaka de-me-ikara.

11. It is often erroneously thought that mobility (naidi) is the converse ofkavu’a (in the sense of “stationary”). In fact, mobility is simply one possi-ble symptom of the items in question failing to constitute a cohesive set.See M. Koppel, “Inclusion and Exclusion” (Hebrew), Higayon 1, pp. 9-11; M. Koppel, “Further Comments on Rov and Kavu’a” (Hebrew),Higayon 4, pp. 49-52; L. Moscovitz, “On the Principles of Majority andIthazek Issura” (Hebrew), Higayon 4, pp. 18-48.

12. For the same reason, if some—but not all—of the objects in the mixtureare mixed with other, permitted objects, the resulting mixture is permitted.While this is not the only possible interpretation of the Gemara inZevahim, it is the one offered by the Rambam. Nevertheless, with regardto a seemingly analogous case considered by the Gemara, a mixture involv-ing valuable objects, Rambam (Ma’akhalot Asurot 16:10) does not permitthe second mixture. It has been speculated that in the latter case, wherebittul fails due to a set property rather than a property of the forbiddenobject, every object in the set is regarded as forbidden so that the presenceof any such object in the second set is sufficient for ithazek issura.

13. Of course, it is possible in such cases that the set from which the object inquestion originates includes an equal number of objects that are P as arenot-P. Such cases are regarded as safek, more about which below.

14. On this understanding bittul applies to the entire set and is roughly analo-gous to the case of votes in Sanhedrin from which it is (by some accounts)learned. This reading of bittul is also neatly parallel with our understandingof kavu’a, which also applies to sets. According to this view, all elements ofa mixed set on which bittul has been effected (in favor of permissibility)should be permitted even simultaneously and to a single person. In fact,Rosh (Hullin 7:37) so rules; Tosafot Rid (Bava Batra 31b), however, dis-agrees. It may be that Tosafot Rid holds that bittul does not apply to theentire set but rather to each individual object in the set separately, alongthe lines of parish. See Kovets Shiurim, Bava Batra, paragraph 127.

15. This reflects the view of Rashba (Hullin 92a) that kavu’a applies any timebittul fails. However, Tosafot (Zevachim 73b s.v. ela, Gittin 64a, s.v. asur)hold that the only authentic cases of kavu’a are nikar bimkomo. All agree,though, that when the failure of bittul is mi-de-rabbanan, as it always is incases like hashivi, that kavu’a also holds only mi-de-rabbanan.

16. The term used in the Gemara is ikba issura. Rambam, however, uses theterm kavu’a. Nevertheless, one should not extrapolate too freely from


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TRADITION

asham taluy to kavu’a in general. Some cases regarded as kavu’a for pur-poses of asham taluy, would not ordinarily be considered cases of kavu’a.For example, if a melakha is performed, possibly on Shabbat, possibly onanother day, an asham taluy may be brought.

17. Recall, though, that RDIK resolves uncertainty but does not dispel it. Onthis basis, Shev Shema’at’ta (2:15) argues that certain talmudic referencesto safek, such as safek asiri (Bava Metsia 7a) and safek mamzer (Kiddushin73a), include cases of RDIK. This claim serves as the starting point for agreat deal of “yeshivishe Torah” but is not essential for an understanding ofthe underlying issues.

18. Recall that we saw that RDLK comes in at least three varieties (dependingon the naturalness of the reference class and the strength of the statisticallaw), the weakest of which is a simple default rule. We can think of safekha-ragil as a degenerate case of RDLK in which the reference class is notdeemed convincing enough to even rate as a default rule. One differencebetween RDLK as a default rule and safek ha-ragil is that RDLK as defaultneutralizes a hazaka de-me-ikara (Yevamot 119a) while the fact that a safekis safek ha-ragil is irrelevant in the face of such a hazaka. Note, though,that at least on one occasion (Ketubot 9a, s.v. ve-i ba’it), Tosafot conflateRDLK as default with safek ha-ragil. Moreover, according to Tosafot, thefact that a safek is ragil is not entirely irrelevant: Tosafot (Pesahim 9a, s.v.ve-im timtsi lomar) hold that safek ha-ragil might not be subject to therule that safek tum’a bi-reshut ha-yahid tamei.

19. One disadvantage of this approach is that sometimes the phrase sefek sefekais used with regard to a mixed set which then falls into another set (forexample, Zevahim 74a). According to this second interpretation of sefeksefeka, the overlap in terminology is misleading and such mixture withinmixture cases must be analyzed in the context of the rules governingmixed sets discussed above. According to the first approach, the conflationof ordinary kinds of sefek sefeka and mixtures within mixtures makes sense:in each the “outer” safek/mixture is treated leniently because the hamurside fails to satisfy the requirement of ithazek issura (see the discussion nearfootnote 12 above).

20. For example, the question of whether a particular woman is more likely tohave had relations subsequent to or prior to her betrothal would appear todepend on the circumstances of the case in question; there is no evidence ofany general statistical rule that the rabbis applied to all such cases. For moreconvincing examples of the inapplicability of RDLK, see Taharot 6:4 andNidda 33b. It is important to note that there are cases of sefek sefeka andpseudo-sefek sefeka which happen to also be RDLK. For example, in Hullin77b, Rashi s.v. ve-khol ha-yoledot (perhaps based on J. Yevamot 16:1) formu-lates the Gemara’s claim that most births are not both living and males interms of sefek sefeka even though stillbirths are clearly a minority. The reasonthat this works is that it happens to also be a RDLK (See Shev Shema’at’ta1:18). Similarly, Rambam (Para Aduma 9:16) employs the terminology ofsefek sefeka to explain the Gemara’s claim (Hullin 9b) that mei hatat thatare found uncovered can be assumed not to have been handled by an agentwho was both human and tamei. This would contradict the law that sefek


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sefeka bi-reshut ha-yahid tamei except that it also happens to be a RDLK.See also Bava Kamma 11a, Tosafot s.v. de-eno.

21. Still, it must be noted that a majority of possibilities as in sefek sefeka is notalways treated in exactly the same way as a majority of concrete objects asin the usual case of RDIK. In Tosefta Taharot 6:2 (as explained in Ketubot15a), we find that in a case of safek tum’a involving parish (the word usedin the Tosefta is nimtsa), we follow the majority even for leniency; only in acase of kavu’a, do we follow the rule that safek tum’a bi-reshut ha-yahidtamei. Nevertheless, in Taharot 6:4 we find that sefek sefeka regardingtum’a bi-reshut ha-yahid is an inadequate basis for leniency. On the face ofit, this would suggest that sefek sefeka is treated differently than RDIK. SeeAhiezer [Yoreh Deah] 2:13. It would appear more appropriate, however, toascribe this anomaly to the generally exceptional nature of the principlesafek tum’a bi-reshut ha-yahid tamei. As is evident from the Tosefta, safektum’a refers specifically to cases of kavu’a, which, as we have seen, is anatypical use of the term safek. Thus, it would appear that the sefek sefekacases in the Mishna are regarded as more similar to the Tosefta’s kavu’a casethan to the RDIK case. In fact, R. Soloveichik argues that the rule thatsafek tum’a bi-reshut ha-yahid tamei employed in the Mishna applies onlywhen the presence of tum’a is certain so that the only uncertainty involveswhether it was metamei other things. Thus all cases of safek tum’a for whichthe rule is relevant may fall under some broader definition of kavu’a (allud-ed to in footnote 16 above with regard to asham taluy). Consequently, therule is inapplicable in the nimtsa case of the Tosefta in which RDIK isapplied to a single object which might not be tamei at all. (It is difficult,however, to reconcile this restriction to the rule with the language of theMishna where the rule is applied: “safek hayeta sham, safek lo hayeta sham.”)See Shiurei ha-Rav Aharon Lichtenstein on Taharot, pp. 167-171. (Foranother example of the atypicality of the rule safek tum’a bi-reshut ha-yahidtamei, see footnote 18 above.)

22. A number of interesting examples of this phenomenon, all beyond thescope of this paper, are subsumed under the name “Bertrand’s Paradox.”See von Mises, p. 77.

23. Halikhot Olam, Agur and Rivash attribute this view to Tosafot but withoutciting a source. Indeed it is not found in the standard Tosafot but can befound in Tosafot Yeshanim, Ketubot 9b (cited in the outer margin in theVilna Shas). The Shakh on Yoreh De’a 110:9, Gloss 63, points out that inmany known cases of sefek sefeka the independence condition fails and thatin fact we require only partial independence—namely, that where there is anatural ordering on A and B (say, A is logically prior to B), it is sufficientthat the question of B or not-B not be resolved given that A but not viceversa. Thus, for example, we find (Taharot 6:4) a sefek sefeka in which thetwo conditions are (A) that one entered a room in which there is a tameiobject and (B) that one touched the object. Obviously, if B holds then Aholds as well. Still, since the question regarding A arises prior to thatregarding B, this is a valid sefek sefeka (albeit an ineffective one for unrelat-ed reasons). In this case, there are not four possibilities, as in the standardsefek sefeka, but rather three (the possibility that one touched the object


26

TRADITION

but was not in the room is eliminated). Still, there is a majority of two outof three.

24. See the discussion in Rabinovitch, chapter 5.25. A number of arcane exceptions are considered by later commentators. A

summary can be found in E. Cohen, “Introduction to Sefek Sefeka”(Hebrew), Sinai 112 (5753), pp. 273-282.

26. See P. Schiffman, “Safek in Halakha and in Law” (Hebrew), Shenaton ha-Mishpat ha-Ivri 1 (5734), pp. 328-352.


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R U : A U VERVIEW OF RABBINIC METHODSu.cs.biu.ac.il/~koppel/rov-traditionversion.pdf · 2006. 9. 29. · Moshe Koppel 1 Moshe Koppel teaches computer science at Bar-Ilan University